Integrand size = 8, antiderivative size = 27 \[ \int \sin ^3(a+b x) \, dx=-\frac {\cos (a+b x)}{b}+\frac {\cos ^3(a+b x)}{3 b} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2713} \[ \int \sin ^3(a+b x) \, dx=\frac {\cos ^3(a+b x)}{3 b}-\frac {\cos (a+b x)}{b} \]
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Rule 2713
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {\cos (a+b x)}{b}+\frac {\cos ^3(a+b x)}{3 b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \sin ^3(a+b x) \, dx=-\frac {3 \cos (a+b x)}{4 b}+\frac {\cos (3 (a+b x))}{12 b} \]
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Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(-\frac {\left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3 b}\) | \(22\) |
default | \(-\frac {\left (2+\sin ^{2}\left (b x +a \right )\right ) \cos \left (b x +a \right )}{3 b}\) | \(22\) |
parallelrisch | \(\frac {-8-9 \cos \left (b x +a \right )+\cos \left (3 b x +3 a \right )}{12 b}\) | \(25\) |
risch | \(-\frac {3 \cos \left (b x +a \right )}{4 b}+\frac {\cos \left (3 b x +3 a \right )}{12 b}\) | \(27\) |
norman | \(\frac {-\frac {4 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {4}{3 b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{3}}\) | \(39\) |
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none
Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \sin ^3(a+b x) \, dx=\frac {\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \, b} \]
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Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \sin ^3(a+b x) \, dx=\begin {cases} - \frac {\sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 \cos ^{3}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \sin ^3(a+b x) \, dx=\frac {\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )}{3 \, b} \]
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Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \sin ^3(a+b x) \, dx=\frac {\cos \left (b x + a\right )^{3}}{3 \, b} - \frac {\cos \left (b x + a\right )}{b} \]
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Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \sin ^3(a+b x) \, dx=-\frac {3\,\cos \left (a+b\,x\right )-{\cos \left (a+b\,x\right )}^3}{3\,b} \]
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